weather derivatives and stochastic modelling of temperature

Hindawi Publishing CorporationInternational Journal of Stochastic AnalysisVolume 2011, Article ID 576791, 21 pagesdoi:10.1155/2011/576791

Research ArticleWeather Derivatives and StochasticModelling of Temperature

Fred Espen Benth1, 2 and Jurate Saltyte Benth3, 4

1 Center of Mathematics for Applications, University of Oslo, P.O. Box 1053 Blindern, 0316 Oslo, Norway2 School of Management, University of Agder, Serviceboks 422, 4604 Kristiansand, Norway3 HØKH, Research Centre, Akershus University Hospital, Lørenskog, Norway4 Faculty Division Akershus University Hospital, University of Oslo, Oslo, Norway

Correspondence should be addressed to Fred Espen Benth, [email protected]

Received 31 January 2011; Revised 4 May 2011; Accepted 17 May 2011

Academic Editor: Maria J. Lopez-Herrero

Copyright q 2011 F. E. Benth and J. Saltyte Benth. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We propose a continuous-time autoregressive model for the temperature dynamics with volatilitybeing the product of a seasonal function and a stochastic process. We use the Barndorff-Nielsenand Shephard model for the stochastic volatility. The proposed temperature dynamics is flexibleenough to model temperature data accurately, and at the same time being analytically tractable.Futures prices for commonly traded contracts at the Chicago Mercantile Exchange on indices likecooling- and heating-degree days and cumulative average temperatures are computed, as well asoption prices on them.

1. Introduction

Protection against undesirable weather events, like, for instance, hurricanes and droughts,has been offered by insurance companies. In the recent decades, the securitization ofsuch weather insurances has taken place, and nowadays one can trade weather derivativecontracts at the Chicago Mercantile Exchange (CME), or more specialized weather contractsin the OTC market. At the CME, one finds derivatives written on temperature and snowfallindices, measured at different locations worldwide. The market for weather derivatives isemerging and gaining importance as a tool for hedging weather risk.

In this paper we propose a new model for the dynamics of temperature whichforms the basis for pricing weather derivatives. The model generalizes the continuous-timeautoregressive models suggested in Benth et al. [1] (see also Dornier and Querel [2] andBenth et al. [3]), to allow for stochastic volatility effects. Signs of stochastic volatility in thetemperature variations have been detected in many data studies, for example, in an extensivestudy of daily US temperature series by Campbell and Diebold [4] and in Norwegian data

2 International Journal of Stochastic Analysis

by Benth and Saltyte-Benth [5]. Campbell and Diebold suggested a seasonal generalizedautoregressive conditional heteroskedasticity (GARCH)model to explain their observations,while here in this paper we suggest to model the variations by the stochastic volatility modelof Barndorff-Nielsen and Shephard [6].

As it turns out, one may derive reasonably explicit expressions for futures prices ofcommonly traded contracts at the CME. Our proposed model is flexible in capturing thestylized features of temperature data, as well as being analytically tractable. To account forseasonality, we introduce a multiplicative structure, which is much simpler to analyse thanthe complex model suggested by Campbell and Diebold [4], where the seasonality comes inadditively in the GARCH dynamics. One of the main advantages with our model is thatit is set in a continuous-time framework, accommodating for a simple application of thetheory of derivatives pricing. For example, we find that the volatility of futures prices onthe cumulative average temperature is given by the temperature volatility, modified by aSamuelson effect inherited from the autoregressive structure of the temperature dynamics.

The risk premium in derivatives prices is parametrized by a time-dependent marketprice of risk. In addition, we include a market price of volatility risk. Thus, the risk loadingin derivatives prices, interpreted as the risk premium in the financial setting, includes bothtemperature and volatility. In mathematical terms, the pricing measure is obtained by acombination of a Girsanov and Esscher transform.

We discuss the estimation of the proposed temperaturemodel on data from Stockholm,Sweden. The purpose of the study is not to give a full-blown fitting of the model, but tooutline the steps and to justify the model. Next, we discuss some issues related to the meanreversion of themodel, where in particular we derive the so-called half-life of our temperaturemodel. The importance of the various terms in the model is discussed.

We present our findings as follows. In the next section we review the basics of theCMEmarket for weather derivatives. Next, in Section 3, our temperaturemodel with seasonalstochastic volatility is defined and analysed. Section 4 contains results on the futures pricedynamics for some commonly traded products at the CME. Some empirical considerations ofour model and its applications to weather derivatives are presented in Section 5.

2. The Market for Temperature Derivatives

The organizedmarket for temperature derivatives at the CME offers trade in futures contracts“delivering” various temperature indices measured at different locations in the world,including the US, Canada, Australia, Japan, and some countries in Europe. In addition, thereare contracts written on snowfall in New York and frost conditions at Schiphol airport inAmsterdam. On the exchange one may also buy plain vanilla European call and put optionson the futures.

The main temperature indices are so-called cooling-degree days (CDDs), heating-degreedays (HDDs), and cumulative average temperature (CAT). The CAT index is for Europeancities mainly. For a measurement period [T1, T2], T1 < T2, the CDD index is measuring thedemand for cooling and is defined as the cumulative amount of degrees above a threshold c.Mathematically we express it as

T2∑

t=T1

max(T(t) − c, 0), (2.1)

International Journal of Stochastic Analysis 3

with T(t) being the daily average temperature on day t, and the average is computed as themean of the daily maximum and minimum observations. The threshold c is in the marketgiven as 18◦C, or 65◦F and is the trigger point for when air-conditioning is switched on. Themeasurement periods are set to weeks, months, or seasons consisting of two or more months,within the warmer parts of the year. A futures written on the CDD pays out an amountof money to the buyer proportional to the index, in US dollars for the American market,and in Euro for the European one (except London, where British Pounds is the currency).By entering such a contract, one essentially exchanges a fixed CDD against a floating, andthereby one may view these futures as swaps.

An HDD index is defined similarly to the CDD as the cumulative amount oftemperatures below a threshold c over a measurement period [T1, T2], that is,

T2∑

t=T1

max(c − T(t), 0). (2.2)

The index reflects the demand for heating at a certain location, and measurement periods aretypically weeks, months, or seasons in the cold period (winter).

The CAT index is simply the accumulated average temperature over the measurementperiod defined as

T2∑

t=T1

T(t). (2.3)

This index is used for European and Canadian cities at CME. CME also operates with adaily average index called the Pacific Rim measured in several Japanese cities. The PacificRim is simply the average of daily temperatures over a measurement period, with the dailytemperature defined as the mean of the hourly observed temperatures.

Temperature futures may be used by energy producers to hedge their demand risk(see Perez-Gonzales and Yun [7] for a study of weather hedging of US energy utilities).Other typical users of such futures contracts for hedging purposes are electricity retailers,holiday resorts, producers of soft drinks and beer, or even organisers of outdoor sportevents, and fairs. Also investors have seen the potential in weather derivatives as a tool forportfolio diversification, since the derivatives are not expected to correlate significantly withthe financial markets (see Brockett et al. [8]).

For mathematical convenience, we will use integration rather than summation anddefine the three indices CDD, HDD, and CAT over a measurement period [T1, T2] by

CDD(T1, T2) =∫T2

T1

max(T(t) − c, 0)dt, (2.4)

HDD(T1, T2) =∫T2

T1

max(c − T(t), 0)dt, (2.5)

CAT(T1, T2) =∫T2

T1

T(t)dt, (2.6)

respectively.


In this paper we are concerned with modelling the temperature dynamics T(t)accurately in continuous time. Moreover, we aim at deriving futures prices based onour proposed dynamics. In the literature, one finds several streams for pricing weatherderivatives. The main approach is to model the stochastic temperature evolution and priceby conditional risk-adjusted expectation (see Benth et al. [3]). This is motivated from fixed-income theory, where zero-coupon bonds are priced using a model for the short rate ofinterest and conditional risk-adjusted expectation (see, e.g., Duffie [9]). Another popular wayto price derivatives is using the so-called burn analysis (see Jewson and Brix [10]). There, onecomputes the historical average of the index in question and prices by this figure. One mayadd a risk loading to this price. A more sophisticated pricing method is the so-called indexmodelling approach (see Jewson and Brix [10]), where one is modelling the historical indexvalue by a probability distribution. Index modelling was applied by Dorfleitner andWimmer[11] in an extensive empirical study of US futures prices at the CME. The drawbackwith thesetwo ways of pricing is that you do not get any dynamics for the futures price. If one wantsto derive option prices, one must have accessible the volatility of the futures price as wellas the price itself. Although applying the burn analysis (index modelling) gives an accurateestimate on the historical index value (distribution), it is expected that a precise model of thetemperature dynamics explains close to equally well the historical average (distribution). Inaddition, one obtains the stochasticity of the temperature evolution, which enables us to findthe futures price dynamics and to price options on such futures.

Brockett et al. [8] applies the indifference pricing approach in a mean-variance settingto valuate weather derivatives. Davis [12] introduces the marginal utility approach to pricingof weather derivatives, which is based on hedging in correlated assets and modelling ofthe index as a geometric Brownian motion. Platen and West [13] suggest using a worldindex as a benchmark for weather derivatives pricing. Equilibrium theory has also been apopular approach for pricing weather derivatives. Cao andWei [14] apply this to analyse UStemperature futures (see also Hamisultane [15] for a more recent study).

3. A Model for the Temperature Dynamics withSeasonal Stochastic Volatility

Let (Ω,F, {F}0≤t≤τ , P) be a complete filtered probability space with τ < ∞ being a finite timehorizon for which all the relevant financial assets in question exist.

We suppose that the temperature T(t) at time t ≥ 0 is defined as

T(t) = Λ(t) + Y (t), (3.1)

where Λ(t) is the deterministic seasonal mean function and Y (t) is a stochastic processmodelling the random fluctuations around the mean. In other words, Y (t) = T(t) − Λ(t) isthe deseasonalized temperatures.

The seasonal mean function Λ(t) is assumed to be real-valued continuous function.To capture the seasonal variations due to cold and warm periods of the year, and possibleincrease in temperatures due to global warming and urbanization, say, a possible structure ofΛ(t) could be

Λ(t) = a + bt + c sin(2π(t − d)

365

), (3.2)


for constants a, b, c, and d, and with time t measured in days. Such a specification of theseasonality function was suggested by Alaton et al. [16] in their analysis of temperaturesobserved in Bromma, Sweden. Benth and Saltyte Benth [5] applied this seasonality functionwhen modelling Norwegian temperature data and later used it for data measured inLithuania and Sweden in the papers [1, 17, 18]. Mraoua and Bari [19] apply such a seasonalfunction on Moroccan data. Other, more sophisticated seasonal functions may of course beused (see Hardle and Lopez Cabrera [20] and Hardle et al. [21]).

Empirical analysis of temperature data in Norway, Sweden, and Lithuania (see Benthet al. [1, 5, 17, 18]), and US and Asian cities (see Cao and Wei [14] and Hardle and LopezCabrera [20]), suggest that the deseasonalized temperatures Y (t) have an autoregressivestructure on a daily scale. This motivates the use of a continuous-time autoregressive (CAR)model for Y (t) (see Brockwell [22] for a general account on such processes).

For p ≥ 1, a CAR(p)-dynamics is defined as follows. Define the p × p-matrix A by

A =

[0 I

−αp · · · −α1

], (3.3)

with αi being positive constants for i = 1, . . . , p, and I the (p − 1) × (p − 1) identity matrix.Observe that the determinant of A is equal to αp > 0, and hence A is invertible. For aone-dimensional Brownian motion B(t), we define the p-dimensional Ornstein-Uhlenbeckprocess X(t) as

dX(t) = AX(t)dt + σepdB(t), (3.4)

where ei, i = 1, . . . , p is the canonical ith unit vector in Rp and σ > 0 constant. A CAR(p)

process is defined as Y (t) = e′1X(t), with x′ being the transpose of a vector x. It seems thata CAR(3) is the most reasonable choice from an empirical point of view (see, e.g., Benthet al. [1] or Hardle and Lopez Cabrera [20], where the volatility σ is allowed to be time-dependent to allow for seasonality effects). In passing we note that a similar model has beenused for wind speed modelling for New York and Lithuania, resulting in a CAR(4) processfor deseasonalized data as the statistically optimal choice (see Saltyte Benth and Benth [23]and Saltyte Benth and Saltyte [24]).

In this paper we are concerned with an extension of the CAR(p) model defined (3.4)to explain seasonality and stochastic volatility effects in the temperature variations. In Benthand Saltyte Benth [5], it was observed that the residuals of temperature after explainingthe autoregressive effects had a clear seasonality and signs of stochastic volatility (from astudy of Norwegian temperatures). This seasonal structure can be partly explained by aseasonal variance. However, one is still left with signs of stochastic volatility observed as anexponentially decaying autocorrelation function for squared residuals. Similar observationswere made by Campbell and Diebold [4] on US data. They suggest a GARCH volatilitystructure where a seasonal level component is added to model this. However, from amodelling and empirical point of view, it seems much more reasonable to consider amultiplicative structure between the seasonality and the stochasticity of the volatility. It isa rather standard approach in time series analysis to apply multiplicative models whendealing with phenomena having positive values. Using a multiplicative structure makes iteasy to preserve the natural condition of positivity of the overall volatility. Also, empiricallyit becomes simple to estimate the seasonality and the stochastic volatility contributions in


a stepwise procedure. We remark in passing that the seasonal GARCH model of Campbelland Diebold [4] may lead to negative values of the volatility. Our proposed stochasticvolatility model will lend itself to analytical tractability when it comes to pricing of weatherderivatives, which seems to become very complicated when using the seasonal GARCHmodel of Campbell and Diebold [4].

Hence, we propose the following CAR(p) model with seasonal stochastic volatility.Suppose that X(t) follows the dynamics

dX(t) = AX(t)dt + φ(t)epdB(t), (3.5)

where

φ(t) = ζ(t) × σ(t), (3.6)

for a bounded positive continuous function ζ(t), and σ(t) being a stochastic process assumedto the positive. We assume that ζ is strictly bounded away from zero, that is, there exists aconstant δ > 0 such that ζ(t) ≥ δ for all t ≤ τ .

In Benth et al. [1], the model (3.5) was considered with σ(t) = 1, that is, with nostochastic volatility effects. The function ζ(t)was estimated on data from Stockholm, Sweden,to be a truncated Fourier series of order four, having a yearly seasonality, that is, ζ(t+k×365) =ζ(t) for k = 1, 2, . . ., t ≥ 0, with time being measured in days. This seasonal volatility has beenobserved and modelled for data in Sweden and Lithuania (see Benth et al. [1, 18]), and USand Asian cities (see Cao and Wei [14], Hardle and Lopez Cabrera [20] and Benth et al.[25]). Recently, Hardle et al. [21] proposed and applied a powerful local adaptive modellingapproach to optimally estimate the volatility.

A class of stochastic volatility processes providing a great deal of flexibility in precisemodelling of residual characteristics is given by the Barndorff-Nielsen and Shephard [6](BNS)model. In mathematical terms, we have

σ2(t) � V (t), (3.7)

with

dV (t) = −λV (t)dt + dL(t). (3.8)

Here, λ > 0 is a constant measuring the speed of mean reversion for the volatility processV (t), which reverts to zero. The process L(t) is assumed to be a subordinator independent ofB, the Brownian motion, meaning a Levy process with increasing paths. In this way one isensured that V (t) is positive.

In the BNS model a dependency on λ in the subordinator is introduced. We choosea subordinator U(t) and define L(t) = U(λt), which then again becomes a subordinator.This is convenient when estimating such a stochastic volatility model. In fact, one mayget relatively explicit distributions for the ”deseasonalized” residuals

√V (t)dB(t), which

become conditionally normally distributed, with mean zero and variance V (t). In stationarityof V , this distribution becomes independent of λ, and therefore one may separate modellingof the distribution of these residuals from the dependency structure in the paths. For


this stochastic volatility model, the squared residuals will have an exponentially decayingautocorrelation function, with decay rate λ. By superposition of such V ’s, the autocorrelationfunction may decay as a sum of exponentials. We refer to Barndorff-Nielsen and Shephard[6] for an extensive analysis of this class of stochastic volatility models. We remark that thestochastic BNS volatility does not become a GARCH dynamics in discrete time, but sharessome similar properties.

In the rest of this paper we suppose that the deseasonalized temperature Y (t) is givenby

Y (t) = e′1X(t), (3.9)

where X(t) is defined in (3.5) and σ(t) defined in (3.7) and (3.8).

4. Temperatures Futures Pricing

In this section we derive the futures price dynamics based on the CDD/HDD and CATindices. This will be done with respect to some pricing measure Q specified in a moment.

We use the intrinsic notation TI(T1, T2) for a temperature index over the measurementperiod [T1, T2], where TI = HDD/CDD or CAT. Let FTI(t, T1, T2) be the futures price fora contract returning the index TI(T1, T2). The futures contracts are settled at the end ofmeasurement period, time T2, and the time t value of a long position in the contract is

exp(−r(T2 − t)){TI(T1, T2) − FTI(t, T1, T2)}, (4.1)

where the constant r > 0 is the risk-free interest rate. Thus, if Q is a probability equivalent toP , we can define the futures price as

FTI(t, T1, T2) = EQ[TI(T1, T2) | Ft] (4.2)

by using that the contract is costless to enter and assuming that the futures price is adaptedto the information at time t, Ft.

One may ask why we do not consider stochastic interest rates in this setup. Infact, it would not be any problem to do so, by, for example, assuming a risk-free spotrate r(t). However, there is no reason why this should depend on the temperature, and areasonable model would state independence between r(t) and T(t). Thus, from the propertiesof conditional expectations, we would end up with the same pricing relations as in (4.2).

Note that the pricing measure Q is only equivalent to P and not assumed to be amartingale measure. If temperature would be a tradeable asset, then the no-arbitrage theoryof asset pricing (see, e.g., Duffie [9]) would demand that Q is an equivalent martingalemeasure, that is, a probability equivalent to P under which the discounted temperature isa Q-martingale. Temperature is obviously not a tradeable asset, and the condition on being amartingale measure is not effective.

To restrict the class of pricing probabilities, we consider a deterministic combinationof a Girsanov and Esscher transform. The Girsanov transform introduces a market price of


risk with respect to the Brownian motion, while the Esscher transform will model a marketprice of volatility risk. In mathematical terms, we define the probability Q as

Q = QB ×QL, (4.3)

with

dQB

dP

∣∣∣∣Ft

= exp

(∫ t

0

θB(s)φ(s)

dB(s) − 12

∫ t

0

θ2B(s)φ2(s)

ds

),

dQL

dP

∣∣∣∣Ft

= exp

(∫ t

0θL(s)dL(s) −

∫ t

0ψL(θL(s))ds

),

(4.4)

for θB, θL being two bounded continuous functions. Here, ψL is the log-moment generatingfunction of L, that is,

ψL(θ) := lnE[exp(θL(1))

]. (4.5)

Note that the Novikov condition is satisfied for the Girsanov change of measure sinceσ2(s) ≥ σ2(0) exp(−λs), ζ(t) ≥ δ and θB being bounded. Furthermore, we assume exponentialintegrability of L, that is, there exists a constant k such that

E[exp(kL(1))

]<∞. (4.6)

Then, the Esscher transform is well defined for all functions θL such that sup0≤t≤τ |θL(t)| ≤ k.We restrict our attention to this class of market prices of volatility risk. We find that

dW(t) = dB(t) − θB(t)φ(t)

dt, (4.7)

is a Q-Brownian motion, independent of L. Moreover, following the analysis in Benth et al.[3], it holds that L is an independent increment process under Q, and, in the case where θL isa constant, a Levy process.

There has been some work trying to reveal the existence and structure of the riskpremium, or the market price of risk, for temperature derivatives. The theoretical benchmarkapproach by Platen and West [13] strongly argues against the existence of a risk premiumin weather markets. The econometric study of US futures prices at CME performed byDorfleitner and Wimmer [11] shows that the index modelling approach gives a very goodprediction of prices, pointing towards a zero market price of risk. The results of Cao and Wei[14] is more inconclusive, as they are able to detect a significant (although small)market priceof risk based on their equilibrium pricing approach. More recently, Hamisultane [15] showedusing New York data that the estimates become very unstable within this pricing framework.Hardle and Lopez Cabrera [20] analyse various specifications of the market price of risk in anextensive empirical study of German futures prices. They find signs of a significant marketprice of risk for these contracts. Based on the abovementioned studies, one may choose the


market price of risk θB as zero. As far as we know, there exists no empirical investigationson the volatility risk premium in weather markets. One may suspect that this is zero, but weinclude θL here for generality. Using θb = θL = 0 would correspond to Q = P and be a choicefollowing the indications of the study in Dorfleitner and Wimmer [11].

In the following proposition we compute the CAT futures price.

Proposition 4.1. The CAT futures price at time t for a contract with measurement period [T1, T2],t ≤ T1, is

FCAT(t, T1, T2) =∫T2

T1

Λ(u)du + a(T2 − t, T1 − t)X(t)

+∫T1

t

a(T2 − s, T1 − s)epθB(s)ds

+∫T2

T1

a(T2 − s, 0)epθB(s)ds,

(4.8)

where

a(u, v) = e′1A−1(exp(Au) − exp(Av)

). (4.9)

Proof. We know that the Q-dynamics of X(t) becomes

dX(t) = AX(t)dt + θB(t)epdt + epφ(t)dW(t), (4.10)

for theQ-BrownianmotionW . Thus, from themultidimensional Ito formula, it holds for u ≥ t

X(u) = exp(A(u − t))X(t) +∫u

t

exp(A(u − s))epθB(s)ds +∫u

t

exp(A(u − s))epφ(s)dW(s).

(4.11)

It follows that

EQ[T(u) | Ft] = Λ(u) + EQ

[e′1X(u) | Ft

]

= Λ(u) + e′1 exp(A(u − t))X(t) +∫u

t

e′1 exp(A(u − s))epθB(s)ds

+ EQ

[∫u

t

e′1 exp(A(u − s))epζ(s)σ(s)dW(s) | Ft

].

(4.12)

By conditioning on the σ-algebra generated by the paths of σ(s), 0 ≤ s ≤ t and Ft, and usingthe tower property of conditional expectation, we find that the last term above is equal to zerofrom the independent increment property of W . Finally integrating with respect to u yieldsthe desired result.


Observe that the market price of volatility risk does not enter the CAT futures priceexplicitly, only the market price of risk θB. We derive the dynamics of the CAT futures pricein the following proposition.

Proposition 4.2. The Q-dynamics of the CAT futures price is

dFCAT(t, T1, T2) = a(T2 − t, T1 − t)epφ(t)dW(t), (4.13)

whereas the P -dynamics becomes

dFCAT(t, T1, T2) = −a(T2 − t, T1 − t)epθB(t)dt + a(T2 − t, T1 − t)epφ(t)dB(t), (4.14)

with a(u, v) defined in Proposition 4.1.

Proof. This follows by a straightforward application of the multidimensional Ito Formula,where the observation that FCAT(t, T1, T2) is a Q-martingale simplify the derivationsconsiderably.

As is evident from this dynamics, the CAT futures price will have a seasonal stochasticvolatility φ(t) = ζ(t)σ(t), dampened by the factor a(T2 − t, T1 − t)ep. This factor can be viewedas the integral of e′1 exp(A(s−t))ep over the measurement period [T1, T2]. Observe that for thecase of a CAR(1) process, which is nothing but an ordinary Ornstein-Uhlenbeck process, thisterm collapses into exp(−α1(u − t)), an exponential damping of the volatility of temperatureas a function of ”time-to-maturity” u− t. This is known as the Samuelson effect in commoditymarkets. In the current context, we have a term a(T2 − t, T1)ep which can be interpreted as theaggregation of the Samuelson effect over the measurement period [T1, T2] (see Benth et al.[3] for a thorough analysis on the Samuelson effect for higher-order autoregressive models).The drift in the P -dynamics of FCAT(t, T1, T2) will depend explicitly on the parameter θB(t),defending the name market price of risk.

We next move our attention to CDD futures, whichwill become nonlinearly dependenton the temperature. Thus, we will not get as explicit results on their prices as for the CATfutures. It turns out to be useful to apply Fourier transform techniques (see Carr and Madan[26]).

Recall from Folland [27] that for a function f ∈ L1(R), we have

f(x) =12π

∫

R

f(y)eixydy, (4.15)

where f is the Fourier transform of f defined by

f(y)=∫

R

f(x)e−ixydx. (4.16)

Note the signs in the exponentials here, which are not following the standard definition.Relevant to the analysis of CDD futures, consider the function f(x) = max(x, 0), which

is not in L1(R), but we can dampen it by an exponential function and get the following lemma.


Lemma 4.3. For any ξ > 0, the Fourier transform of the function

fξ(x) = exp(−ξx)max(x, 0), (4.17)

is

fξ(y)=

1(ξ + iy

)2 . (4.18)

Proof. We have

fξ(y)=∫

R

fξ(x)e−ixydx =∫

R

xe−(ξ+iy)xdx. (4.19)

The result follows by straightforward integration.

We apply this in our further analysis of the CDD futures price.

Proposition 4.4. The CDD futures price at time t for a contract with measurement period [T1, T2],t ≤ T1, is

FCDD(t, T1, T2) =12π

∫

R

fξ(y) ∫T2

T1

Ψ(t, s,X(t), σ2(t), y

)dsdy, (4.20)

with fξ(y) given in Lemma 4.3 and Ψ(t, s, x, σ2, y) given by

lnΨ(t, s, x, σ2, y

)=(ξ + iy

)m(t, s, x) +

12(ξ + iy

)2∫s

t

(e′1 exp(A(s − u))ep

)2ζ2(u)e−λ(u−t)du σ2

+∫ s

t

ψL

(12(ξ + iy

)2∫ s

v

(e1 exp(A(s − u))ep

)2ζ2(u)e−λ(u−v)du + θL(v)

)

− ψL(θL(v))dv,(4.21)

where

m(t, s, x) = Λ(s) + e′1 exp(A(s − t))x +∫s

t

e′1 exp(A(s − u))epθB(u)du − c. (4.22)

The constant ξ > 0 can be arbitrarily chosen.

Proof. First, from the definition of the CDD index and the Q-dynamics of X(t) we find

max(T(s) − c, 0) = max(Λ(s) + e′1X(s) − c, 0

)

= max(m(t, s,X(t)) +

∫s

t

g(s − u)φ(u)dW(u), 0),

(4.23)


where we have introduced the short-hand notation g(u) = e′1 exp(Au)ep. From Lemma 4.3we find that (after commuting integration and expectation by appealing to the Fubini-TonelliTheorem)

EQ[max(T(s) − c, 0) | Ft] =12π

∫

R

fξ(y)EQ

[exp

((ξ + iy

)(m(t, s,X(t))

+∫ s

t

g(s − u)ζ(u)σ(u)dW(u)))

| Ft

]dy

=12π

∫

R

fξ(y)exp

((ξ + iy

)m(t, s,X(t))

)

× EQ

[exp

((ξ + iy

) ∫s

t

g(s − u)ζ(u)σ(u)dW(u))

| Ft

]dy .

(4.24)

The last equality follows from the Ft-adaptedness of X(t). To compute the conditionalexpectation in the last expression, we apply the tower property of conditional expectationswith respect to the filtration Fσ

t generated by the paths of σ(u), 0 ≤ u ≤ τ and Ft. This yieldsfrom the independence between σ(t) and B(t) and the independent increment property ofBrownian motion,

EQ

[exp

((ξ + iy

) ∫ s

t


| Ft

]

= EQ

[EQ

[exp

((ξ + iy

) ∫s

t


| Fσt

]| Ft

]

= EQL

[exp

(12(ξ + iy

)2∫s

t

g2(s − u)ζ2(u)σ2(u)du)

| Ft

].

(4.25)

We have

σ2(u) = V (u) = e−λ(u−t)σ2(t) +∫u

t

e−λ(u−v)dL(v). (4.26)

Hence, by using the adaptedness of σ2(t) to Ft and the independent increment property of Lunder QL,

EQL

[exp

(12(ξ + iy

)2∫s

t

g2(s − u)ζ2(u)σ2(u)du)

| Ft

]

= exp(12(ξ + iy

)2∫s

t

g2(s − u)ζ2(u)e−λ(u−t)duσ2(t))

× EQL

[exp

(12(ξ + iy

)2∫s

t

g2(s − u)ζ2(u)∫u

t

e−λ(u−v)dL(v)du)]

.

(4.27)


Invoking the stochastic Fubini theorem (see Protter [28]) gives

∫s

t

g2(s − u)ζ2(u)∫u

t

e−λ(u−v) dL(v)du =∫s

t

∫s

v

g2(s − u)ζ2(u)e−λ(u−v)dudL(v). (4.28)

Finally, we compute the expectation by using the Radon-Nikodym derivative of QL withrespect to P and the log-moment generating function of L. This proves the proposition.

Although seemingly very complex, the price dynamics of a CDD futures can besimulated by using the fast Fourier transform (FFT) technique (see Carr and Madan [26]for a discussion of this in the context of options). In fact, given X(t) and σ2(t), one usesFFT to compute FCDD(t, T1, T2). This procedure will involve numerical integration over themeasurement period and in the expression for Ψ(t, s,X(t), σ2(t), y). Since FFT is ratherefficient, this seems to be preferable to a direct Monte Carlo simulation of FCDD(t, T1, T2).

The CDD futures price depends explicitly on the current states of the volatility σ2(t)and CAR process X(t). Since we define the deseasonalized temperature as the first coordinateof X(t), this is observable from today’s (timemoment t) temperature after removing the valueof the seasonal functionΛ(t). However, the remaining p−1 states of X(t)must be filtered fromdeseasonalized temperature observations. One may appeal to the Kalman filter for doingthis, although the seasonality and stochastic volatility complicates this procedure. A temptingalternative is to use an Euler discretization of the CAR dynamics as in Benth et al. [3] to relatethe coordinates of X(t) to lagged observations of temperature. (In Benth et al. [3], the Eulerdiscretization is performed for a time continuous, but deterministic, volatility. Note that thereis no change in the steps for developing the same identification when having a stochasticvolatility.) It is easily seen from the definition of X(t) that the quadratic variation process ofthe last coordinate, Xp(t) = e′pX(t), is equal to

d⟨Xp

⟩t= φ2(t)dt, (4.29)

and thus we may apply this to recover σ2(t) from observing the path of Xp(t).Since FCDD(t, T1, T2) depends explicitly on σ2(t), we get the interesting consequence

that even though the temperature dynamics has continuous paths, the CDD futures dynamicswill jumpwhen σ2(t) jumps. Thus, the futures price dynamics will have discontinuous paths.See Benth [29] for a similar result in the context of commodity and power markets using theBNS stochastic volatility model. Note also that the market price of volatility risk appearsexplicitly in the price, contrary to CAT futures.

The HDD futures price dynamics FHDD(t, T1, T2) can be computed analogously to theFCDD(t, T1, T2) price. However, we may also resort to the so-called HDD-CDD parity whichwe collect from Benth et al. [1, Proposition 10.1]. It holds that

FHDD(t, T1, T2) = FCDD(t, T1, T2) + c(T2 − T1) − FCAT(t, T1, T2). (4.30)

We have formulas for both the CDD and CAT futures prices, and thus FHDD(t, T1, T2) readilyfollows from (4.30).


4.1. A Discussion on Pricing of Options on Futures

The technique using Fourier transform is well adapted for option pricing as well, and we willbriefly discuss this in connection with European call and put options written on CAT andCDD futures.

We start with a call option on a CAT futures, with exercise time τ and strike price K,on a futures with measurement period [T1, T2], τ ≤ T1. The price of this call option at timet ≤ τ is given by the no-arbitrage theory as

CCAT(t, τ, T1, T2) = e−r(τ−t)EQ[max(FCAT(τ, T1, T2) −K, 0) | Ft]. (4.31)

As it turns out, the derivation of CCAT using Fourier techniques follows more or less exactlythe same steps as computing the CDD futures price. In fact, we observe from Proposition 4.1,that

FCAT(τ, T1, T2) = FCAT(t, T1, T2) +∫ τ

t

a(T2 − s, T1 − s)epφ(s)dW(s). (4.32)

Moreover, the payoff function of the call option is the same as the CDD index on a given day.By conditioning on the paths of σ(t), we can compute the option price step by step as in theproof of Proposition 4.4. This results in the following proposition.

Proposition 4.5. The price at time t of a call option on a CAT futures with measurement period[T1, T2], exercise time τ ≥ t, τ ≤ T1, and strike K, is

CCAT(t, τ, T1, T2) =12π

∫

R

fξ(y)Φ(t, τ, FCAT(t, T1, T2), σ2(t), y

)dy, (4.33)

with fξ(y) given in Lemma 4.3 and

lnΦ(t, τ, F, σ2, y

)=(ξ + iy

)(F −K) +

12(ξ + iy

)2

×∫ τ

t

(a(T2 − s, T1 − s)ep

)2ζ2(s)e−λ(s−t)dsσ2

+∫ τ

t

ψL

(12(ξ + iy

)2∫ τ

u

(a(T2 − s, T1 − s)ep

)2ζ2(s)e−λ(s−u)ds + θL(u)

)

− ψL(θL(u))du,(4.34)

and a(u, v) defined in Proposition 4.1.

Options on CDD futures are far more technically complicated to valuate. However,we may here as well resort to Fourier techniques and in principle obtain transparent priceformulas. Considering a call option at time t, with strike price K and exercise time τ ≥ t,


we have the price as in (4.31), except that we substitute ”CAT” with ”CDD.” Recalling fromProposition 4.4, we have

FCDD(τ, T1, T2) =12π

∫

R

fξ(y) ∫T2

T1

Ψ(τ, s,X(τ), σ2(τ), y

)dsdy. (4.35)

Furthermore, from the dynamics of X(s) and σ2(s), we can express both in terms of theircurrent states X(t) and σ2(t), in addition to integrals with respect to W and L from t to τ .Similar considerations as for the CAT futures option can then be made, however, leading to atechnically complex formula. We leave the details to the reader.

5. Some Empirical Considerations

We refer to an empirical estimation which can be found in Benth et al. [3] as a startingpoint for our discussion. There a data set of daily average temperatures measured in degreesCelsius in Stockholm, Sweden, is studied. The data are recorded from January 1, 1961 toMay 25, 2006, altogether 16,570 data points after observations on February 29 in every leapyear were removed. The parameters of the seasonal function Λ(t) as defined in (3.2) wereestimated by least squares approach to be a = 6.37, b = 0.0001, c = 10.44, and d = −161.17. Thismeans that there is a small increase in average temperature over the considered measurementperiod. The overall average temperature is around 6.3◦C. The amplitude is ±10.44, whichmeans that on average in the winter temperatures go down to around −4◦C, while summertemperatures reach above 16.7◦C. After removing the seasonal function from the data, onefinds based on the autocorrelation structure of the residuals that a CAR(3) process fits thedeseasonalized data very well. The estimated parameters in the A matrix are α1 = 2.04,α2 = 1.34 and α3 = 0.18, leading to eigenvalues with negative real part and thus A leadsto a stationary CAR dynamics.

Our main concern in this paper is the precise modelling of the residuals. We haveproposed a model where the volatility φ(t) is defined as the product of a deterministicseasonal function ζ(t) and a stochastic volatility process σ(t). The natural path to follow inestimating this is to first estimate the seasonal function. We use the approach suggested inBenth and Saltyte Benth [5, 17], where a Fourier series of order four is chosen for ζ2(t), andestimated on daily empirical variances. Next, the residual data are divided by the estimatedζ(t) function, in order to deseasonalize them.

The residuals after removing the effect of the function ζ(t) and the autoregressiveeffects, are close to normally distributed (see Figure 10.11 in Benth et al. [3]). However, thereare signs of heteroskedasticity, in particular, a normality plot shows signs of a heavy left tailin the distribution (see Figure 10.12 in Benth et al. [3]). Also, the P value of 0.002 of theKolmogorov-Smirnov test suggests to reject the normality hypothesis of the residuals. Weremark in passing that recently Hardle et al. [21] have proposed a local adaptive approachto find an optimal smoothing parameter to locally estimate the seasonality and volatility. Theapproach aims at correcting the nonnormalities in the residuals. Their approach improves thenormal distribution fit to residuals formany cities, but there are still signs of heavy tails (see inparticular the QQ-plot of Berlin data in Figure 11 of Hardle et al. [21]). We also remark that inBenth and Saltyte Benth [5], the empirical seasonal volatility was used as the estimate of ζ(t),


0 5 10 15 20−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

Lag

LN

ofACFof

squa

redresidua

ls

Figure 1: The empirical autocorrelation function on a log-scale for de-seasonalized squared residualstogether with a fitted line for the first 10 lags.

but still a clear sign of stochastic volatility were present when analysing the autocorrelationfunction of the squared residuals.

In Figure 1 we show the autocorrelation function of the squares of the deseasonalizedresiduals on a log-scale. Characteristically, the autocorrelation function decays with the lags,and eventually wiggle around zero. As mentioned earlier, the stationary autocorrelationfunction of σ2(t) = V (t) is an exponential function with decay rate equal to the speed of meanreversion λ of V (t), meaning a linearly decaying autocorrelation function on a log scale. Wefit a linear function to the first 10 lags, with the estimate λ = 0.21. In Figure 1 the estimatedline is shown together with the empirical logarithmic autocorrelation function.

Admittedly, the linear fit to the log-autocorrelation function in Figure 1 is very rough.Since the line does not intersect at zero, it cannot be related directly to an exponentialautocorrelation function. One may mend this deviancy by considering a sum of exponentials.From Figure 1 wemay assume a very steeply decaying autocorrelation function for small lags(up to 2), and thereafter decaying according the fitted line shown in the plot. Even more, afterapproximately 10 lags the autocorrelation function seems to flatten out, and a new line witha different slope would fit better. This could easily be incorporated into our framework byconsidering a superpositioning of three processes of the type V (t) modelling the volatility.In that case, we would get a theoretical autocorrelation function being the sum of threeexponentials, which would be approximately represented as a piecewise linear function onlog scale if the mean reversion speeds λ1, λ2, and λ3 are sufficiently distinct. This couldbe attributed to a fast reversion of big volatility changes, whereas the medium to smallervariations in volatility are slowly reverting, say. The estimated λ would approximatelycorrespond the the mean reversion of medium volatility changes. In order to estimate sucha superposition, one must resort to filtering techniques, which are discussed in detail inBarndorff-Nielsen and Shephard [6].

The final step in estimating our temperature model is to fit a subordinator process L(t).This is done by appealing to the ingenious parametrization of L(t) by Barndorff-Nielsen andShephard [6]. By using L(t) = U(λt) for a subordinator U in the model for V (t) = σ2(t), theyshow that the stationary distribution of V (t) is independent of λ. Thus, the model for the


residuals will become a variance mixture model, being a conditionally normally distributedwith mean zero and variance equal to the stationary distribution of σ2(t). For example,choosing the stationary distribution of σ2(t) to be generalized inverse Gaussian, the variancemixture model of the residuals becomes generalized hyperbolic distributed (see Barndorff-Nielsen and Shephard [6]) for discussion on this).

Based on this, we first select and fit a distribution F to the deseasonalized residuals,and secondly construct the Levy process L which has the required stationary distributionG. The process L is called the background driving Levy process, and Barndorff-Nielsen andShephard [6] show that the distribution Gmust be self-decomposable in order for L to exist.In Benth and Saltyte Benth [5], a generalized hyperbolic distribution was successfully fittedto the deseasonalized temperature residuals for data collected in several Norwegian cities. Inthis case, there exists a subordinator L such that the stationary distribution of σ2(t) becomesgeneralized inverse Gaussian. The Levy measure of L can be explicitly characterised in termsof the selected distribution for the residuals.

One may ask whether some or all of the parameters in the specified temperaturemodel may be time dependent. In our empirical analysis of Stockholm data, we have notdetected any structural changes in the seasonality function Λ(t). Furthermore, in an AR(1)-specification of the temperature model for Stockholm, we have investigated if there are anyseasonality in the speed of mean reversion (study not reported). Looking at the estimates forparticular months over the year, we did not find any pattern defending a seasonality in thespeed of mean reversion (or, in the specification of the AR-matrixA in our context). However,we refer to the paper by Zapranis and Alexandridis [30] for an analysis of temperaturedata in Paris, where the authors detect and model a seasonal mean reversion. We have notinvestigated if any of the other parameters in our model may vary with time.

To understand how fast the temperature dynamics is reverting back to its long-termaverage Λ(t), we discuss the notion of half-life of the stochastic process Y (t). In Clewlow andStrickland [31] the half-life is defined to be the expected time it takes before the process isreturned half way back to its mean from any position. We express this mathematically as thesmallest time τ > 0 so that

E[Y (τ)] =12Y (0). (5.1)

For an Ornstein-Uhlenbeck process (where p = 1, and σ(t) = 1), Clewlow and Strickland [31]derives that τ = ln 2/α. In the next Lemma we derive the half-life of Y (t).

Lemma 5.1. The half-life of Y (t) defined in (3.9) is given by the solution of the equation

e′1 exp(Aτ)e1 =12, (5.2)

when assuming that e′kX(0) = 0 for k = 2, . . . , p.

Proof. First, we have after using the tower property of conditional expectations that

E[Y (τ)] = e′1 exp(Aτ)X(0). (5.3)


Putting equal to Y (0)/2 yields

e′1 exp(Aτ)X(0) =12e′1X(0). (5.4)

Assuming e′kX(0) = 0 for k = 2, . . . , p, we find that X(0) = e1Y (0), and hence the resultfollows.

Note that the first coordinate of X(0) is equal to Y (0) by definition. For convenience,we let the other coordinates be equal to zero in the above result. In reality, we should estimatethe other states of X(0) by filtering the temperature data series in order to reveal theirvalue at time zero. This means that the half-life becomes state dependent for higher-orderautoregressive models.

For a temperature dynamics based on the Stockholm estimates referred to above, wefind the solution to the half-life equation in Lemma 5.1 to be τ = 5.94, that is, it takes onaverage slightly less than six days for today’s temperature to revert half-way back to its long-term level.

Let us investigate the contribution from the various terms in the CAT futures pricedynamics in order to get a feeling for their relative importance in the context of weatherderivatives. For illustrative purposes, we choose a June contract, which starts measurementat time T1 = 151 and ends at time T2 = 181 (supposing time is measured from January 1).Recall the CAT futures price in Proposition 4.1. The first term with the seasonal function isequal to 191.5. Suppose now that current time is one week prior to June 1, that is, t = 144,and let e′

kX(t) = 0 for k = 2, 3, but recalling e′1X(t) = Y (t). We set Y (t) = 5, meaning that

the temperature is 5◦C above the long-term mean level for that particular day. This leadsto a value of 11.8 of the second term, which is around 5% of the value of the mean. Bymoving time t to start of measurement, t = 151, we get the value 37.6 of the second terminstead, which is close to 20% of the contribution of the long-term function to the CAT-futuresprice. This shows that the mean reversion of the temperature dynamics towards the seasonalmean function kills the effect of daily temperature variations quickly, and when being farfrom start of measurement the CAT futures price will be essentially constant (and equal tothe seasonal mean for the measurement period). However, as we get closer to the start ofmeasurement, the effect of the temperature variations become gradually more importantand will impact the CAT futures price significantly. In Dorfleitner and Wimmer [11], anexample of a seasonal futures contract (Chicago HDD winter contract) is presented, wherethe price is constant except from the last week prior to measurement. Then the observedfutures price starts to wiggle. The seasonal function plays a dominating role of setting thelevel of the price. It is worthwhile to emphasize that the length of measurement periodis of importance here. The shorter the measurement period will be, the smaller will thedamping factor a(t, τ1, τ2) become, and the more sensitive the CAT futures price will becometo variations in temperature. This effect is even more emphasized by the fact that the seasonalterm becomes smaller with a smaller measurement period, increasing the relative importanceof the second term. With the introduction of weekly futures contracts at the CME, this isan important observation. Obviously, the two last terms with the market price of risk cancontribute arbitrarily high or little to the CAT futures price, by adjusting the level of θB. In aconcrete application, one would estimate the θB from historical CAT futures prices. Referringto the studies in Benth et al. [25] and Hardle and Lopez Cabrera [20], we find that θB of


the size 0.2 is reasonable. Thus, the terms involving θB will contribute very little compared tothe seasonal function and the second term.

Let us investigate the stochastic volatility contribution to the CDD price. As we seefrom the pricing formula in Proposition 4.4, the influence of the stochastic volatility σ(t)appears in the expression of Ψ(t, s, x, σ2, y), both explicity as σ2(t), and implicitly via thecharacteristic function of the subordinator L driving the dynamics of the volatility process. Ifwe consider a model with no stochastic volatility, it is natural to suppose that σ2(t) = 1. Thefunction Ψ takes the form

lnΨconst(t, s, x, y

)=(ξ + iy

)m(t, s, x) +

12(ξ + iy

)2∫ s

t

(e′1 exp(A(s − u))ep

)2ζ2(u)du, (5.5)

withm(t, s, x) defined in Proposition 4.4. Then, we find the difference between the stochasticvolatility and constant cases as

lnΨ(t, s, x, σ2, y

)− lnΨconst

(t, s, x, y

)

=12(ξ + iy

)2∫s

t


)2ζ2(u)du

{σ2(t) − 1

}

+∫s

t

ψL

(12(ξ + iy

)2∫ s

v


)2ζ2(u)e−λ(u−v)du + θL(v)

)

− ψL(θL(v))dv.

(5.6)

The first term is stochastically varying with the volatility around one, while the second termis determined by the market price of volatility risk through the characteristic function of L.If σ2(t) is slowly varying around its mean, which naturally should be equal to one, then thisfirst termwill not contribute significantly. But due to potential jumps in the volatility, we mayexperience values of σ(t) significantly larger than one, and thus impacting the futures price.The contribution of the second term depends on how big the market price of volatility risk is.

We are not aware of any empirical studies of the potential existence of a market priceof volatility risk. Onemay estimate θL in the same way as θB, by for example, inferring it fromminimizing the distance between theoretical and observed futures prices. Motivated by thestudies of the market price of risk mentioned above (see Cao and Wei [14], Dorfleitner andWimmer [11], Hamisultane [15], and Hardle and Lopez Cabrera [20]), one may suspect thatthe market price of volatility risk will be small and maybe not significant. It is to be noted,however, that even small values of θL may be big relative to the volatility, and thus modifysignificantly the effect of stochastic volatility viewed under the pricing measure Q.

In view of the existence of European-style call and put options written on thetemperature futures, precise knowledge of the volatility is important. The volatility of theunderlying temperature futures will determine the price of the option and play a crucialrole in a hedging strategy of the option. In particular, the Samuelson effect of the volatilityof the temperature futures makes the options particularly sensitive to the volatility closeto measurement. Hence, a precise model for the stochastic volatility is important. Thenonlinearity in the payoff of options also makes second-order effects in the underlying


dynamics more pronounced, and thus even small stochastic volatility variations may becomesignificant in the option dynamics.

Acknowledgments

The authors are grateful for the comments and critics from two anonymous referees, whichled to a significant improvement of the original paper. Fred Espen Benth acknowledgesfinancial support from the project Energy markets: modelling, optimization and simulation(emmos), funded by the Norwegian Research Council under Grant 205328/v30.

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